Figure 8.2. SEM backscattered images of Pb0.92Sr0.08Se and Pb0.88Sr0.12Se ingots, with corresponding EDS mapping result in b) and c).
155
Figure 8.3. Infrared absorption spectrums for undoped Pb1-xSrxSe, band gaps were obtained by extrapolating the squared absorption coefficient (not shown) versus photon energy hv.
Optical absorptions in the lead chalcogenides begin with direct transitions across the fundamental gap at the L point. Because these direct transitions do not require phonon participation, they dominate the absorption spectra compared with the L-Σ indirect transitions, which should occur at higher energies. This limits traditional optical absorption to accurately determining the position of Σ band edge.
To obtain such information, we take a look at the Pisarenko relation at 300 K for samples with different Sr content (Figure 8.4), as the change of band structure will consequently affect the transport properties. For PbSe, due to the large offset between two valence-band maxima at room temperature, contribution from the secondary Σ band on transport is negligible and the Pisarenko relation (data in black squares) can be approximated with a single band model (black curve) up to high carrier density of 2 × 1020 cm-‐3 (difference seen from 1 × 1020 in modeling but experimentally undistinguishable). With the addition of Sr, the Seebeck coefficients start to deviate from the curve significantly at high doping levels, with higher values compared to PbSe given the same carrier density. Among the Pb1-xSrxSe alloys the Seebeck coefficient also increases as the Sr content increases. If increased Seebeck values were simply due to a larger effective mass, the Seebeck value would be proportionally larger at all nH, which is not observed in Pb1-xSrxSe alloys with low nH. Alternatively the deviation from a single band model (the black curve) at high nH indicates the contribution from a second band becomes important as the Fermi level moves into the second valence band, which is also the explanation for the similar Pisarenko behavior in p-type PbTe. The continuous change in the direct L-L band gap due to SrSe alloying can explain the gradual change
2% Sr 4% Sr 8% Sr 12% Sr
of Pisarenko relation. If the energy of the L valence band is reduced as the band gap increases and this reduces the band offset in PbSe between L and Σ valence bands, the secondary Σ band will play a noticeable role in heavily doped, Sr containing PbSe. Assuming half of band gap change results in decreasing separation between L and Σ valence bands, as inspired by observation260 on thin films at 77 K, the calculated Pisarenko relations for each Pb1-xSrxSe alloy composition are shown in Figure 8.4 with colored curves, and are in reasonable agreement with the observed results.
The details about the calculation and relevant considerations are length thus will be included in the last section of this chapter.
Figure 8.4. Pisarenko relation of p-type PbSe and Pb1-xSrxSe. Calculated result matches observation reasonably well, suggesting the difference seen in Pb1-xSrxSe comes from reduced gap
between L and Σ valence bands.
For more independent evidence of how the band structure changes with Sr content, we performed first principle calculation (performed by collaborator Yoshiki Takagiwa) using the Korringa-Kohn- Rostoker Green function formalism under the coherent potential approximation263-265 (KKR-CPA).
For such calculation the experimental room-temperature lattice constants and the von Barth–Hedin formula266 for the exchange energy were used. For all atoms (Pb, Se, and Sr), the angular momentum cut-off, lmax = 2, was set and semi-relativistic calculations of core level were employed.
A dense mesh of 550 k points in the irreducible wedge of the Brillouin-zone was used. Final converged total energy below 10-6 Ry was applied in the self-consistent cycle. The KKR-CPA
PbSe2% Sr 4% Sr 8% Sr 12% Sr 12% Sr
8% Sr
4% Sr
2% Sr
157 method is a powerful tool for visualizing the electronic density of states (DOS) for disordered materials without establishing a supercell and is widely used in studies of thermoelectrics with random substitution65, 165, 267, 268. The calculated DOS for each composition (Figure 8.5 a)) shows an abrupt increase around -0.1 eV, this is attributed to the additional states in the Σ band and its position is used to estimate the gap between L and Σ bands. The trend clearly showed the reduction of gap between L and Σ bands. These are in good consistence with the proposed model as well as optical measurements (Figure 8.5 b).
Figure 8.5. a) calculated DOS of Pb1-xSrxSe, inset shows suggested change of band structure with Sr content, which is used to calculation Pisarenko relation. b) Calculated band gap matches with
experimental results, and calculated gap between two valence bands consistent with the model.
Temperature dependent band gap measurement (Figure 8.6) shows that the rate of band gap increase with temperature, dEg/dT, is also decreased as the increase of Sr, and this rate decreases linearly with Sr content up to 12% at about 0.018 meV K-1/Sr%, so the effect of alloying with Sr on band structure, is not merely moving L and Σ bands closer, but also decreased the rate of how fast the L band position change with temperature. When all these effects are taken into account, we could plot out the reduced energy gap between L and Σ for each alloy composition at different temperatures (Figure 8.7). For 2%, 4%, 8% and 12% Sr, the two bands are effectively converged (gap within 3kBT) at 600K, 550K, 400 K and 300K, respectively. Consider that good p-type samples are heavily doped, the Σ would contribute to transport at even lower temperatures for Pb1-xSrxSe alloys.
Eg, experiment Eg, KKR-CPA ΔEL-Σ, model ΔEL-Σ, KKR-CPA
a
Density of States (Arb. Units)
b
Figure 8.6. a) temperature dependent bang gap of Pb1-xSrxSe and b) change in rate dEg/dT with Sr content.
Figure 8.7. Reduced energy gap between two valence bands in Pb1-xSrxSe as function of temperature, dashed line denotes 3kBT.